Introduction

In this document, we will give you several examples of Bayesian data analysis.

Load packages

library(dplyr)
library(tibble)
library(purrr)
library(tidyr)
library(forcats)
library(gtools)
library(patchwork)
library(broom)
library(broom.mixed)
library(modelr)
library(brms)
library(tidybayes)
library(ggdist)
library(bayesplot)
library(ggplot2)
library(knitr)
BRM_BACKEND <- ifelse(require("cmdstanr"), 'cmdstanr', 'rstan')

Dataset

The following dataset is from experiment 2 of “How Relevant are Incidental Power Poses for HCI?” (Jansen & Hornbæk, 2018). Study participants were asked to either make an expansive posture or a constrictive posture before performing a task. The experiment investigated whether posture could potentially have an effect on risk taking behavior.

First, we load the data.

pose_df = readr::read_csv("data/poses_data.csv", show_col_types = FALSE) %>%
  mutate(condition) %>%
  group_by(participant)
head(pose_df %>% select(participant, condition, change))
participant condition change
1 expansive 3.362832
2 constrictive 29.147982
3 expansive 25.409836
4 constrictive 54.069767
5 expansive -36.644592
6 constrictive 29.756098

Let’s plot out data

wrap_plots(
pose_df %>% 
  ggplot(aes(x = change, fill = condition)) +
    geom_histogram(binwidth = 10) +
    coord_cartesian(xlim = c(-50, 200)) +
    scale_color_theme() + 
    theme_density_x,
pose_df %>% 
  ggplot(aes(x = change)) +
    geom_point(aes(y = 0, color = condition), size = 2) +
    coord_cartesian(xlim = c(-50, 200)) +
    scale_color_theme() + 
    theme_density_x,
nrow = 2)

The data has been aggregated for each participant: - condition = expansive indicates expansive posture, and condition = constrictive indicates constrictive posture - The dependent variable is change which indicates the percentage change in risk-taking behavior. Thus, it is a continuous variable.

For the purposes of this demo, we are only concerned with these two variables. We can ignore the other variables for now.

Intuition of Bayesian Statistics

The Bayesian t-test (BEST) assumes that the data in the two conditions arises from two separate t-distributions. In the following section, we will describe the process for one of the conditions in the experiment.

We will use the \(Normal(\mu = 20, \sigma = 20)\) as the prior distribution.

First, we define some functions for manual calculation of the posterior normal distribution:

sigma_post = function(sigma_prior, sigma, n = 1) {
  sqrt(1 / (1 / (sigma_prior^2) + n / (sigma^2)))
}
mu_post = function(mu_prior, sigma_prior, mu, sigma, n = 1) {
  tau = sigma_post(sigma_prior, sigma, n)
  (tau^2 / sigma_prior^2)*mu_prior + (n * tau^2 / sigma^2)*mu
}
d.p2 = tibble(
  group = c("prior", "expansive", "constrictive", "posterior"), 
  mu = c(0, 32.82, 31.61, mu_post(0, 20, 32.82, 7.52)), 
  sd = c(20, 7.52, 7.06, sigma_post(20, 7.52))
) %>%
  mutate(
    cutoff_group = list(c(1:7)),
    cutoff = list(c(0, 15, 25, 30, 32.82, 40, 100))
  ) %>%
  unnest(c(cutoff_group, cutoff)) %>%
  mutate(
    cutoff = if_else(group == 'prior', 100, cutoff),
    x = map(cutoff, ~seq(from = -100, ., by = 0.1)),
    y = pmap(list(x, mu, sd), ~ dnorm(..1, ..2, ..3))
  )

In the plot below, we show the raw data distribution for the two conditions:

p1 = pose_df %>%
  ggplot() +
  geom_point(aes(x = change, y = condition, colour = condition), 
             position = position_jitter(height = 0.1), alpha = 0.7) +
  scale_color_theme() + 
  labs(y = "Condition") +
  theme(
    legend.position = "none", 
    axis.line.y = element_blank(), 
    axis.ticks.y = element_blank(),
    axis.title.y = element_blank()
  ) +
  scale_x_continuous(limits = c(-150, 250), breaks = seq(-150, 250, by = 50))

p2.blank = tibble(y = c("expansive", "constrictive"), x = 0) %>%
  ggplot(aes(x, y)) +
  scale_color_theme() + 
  theme_density

cowplot::plot_grid(p2.blank, p1, nrow = 2)

First, we define a function to help us plot different cutoff groups.

plot_preliminary <- function(data, group_num, group_name, linewidth_in = 1){
  data %>%
  filter(cutoff_group == group_num & group %in% group_name) %>%
  unnest(c(x, y)) %>%
  ggplot(aes(x, y)) +
  #geom_line(aes(color = group), size = 1) +
  # density
  geom_area(aes(fill = group, color = group, alpha = as.character(group)), position = "identity", linewidth = linewidth_in) +
  scale_x_continuous(limits = c(-150, 250)) +
  scale_y_continuous(limits = c(0, 0.1)) +
  scale_alpha_manual(
    values = c(
      'posterior' = .5,
      'prior' = .3,
      'expansive' = .3,
      'constrictive' = .3
    ),
  ) + 
  scale_color_theme() +
  theme_density +
  theme(legend.position = 'none')
}
cowplot::plot_grid(
  plot_preliminary(d.p2, 0, NULL), 
  p1, nrow = 2)

Next, we plot the prior density:

cowplot::plot_grid(
  plot_preliminary(d.p2, 1, 'prior'), 
  p1, nrow = 2)

Then we describe step by step, how the likelihood is computed:

cowplot::plot_grid(
  plot_preliminary(d.p2, 1, 'expansive'), 
  p1, nrow = 2)

cowplot::plot_grid(
  plot_preliminary(d.p2, 2, 'expansive'), 
  p1, nrow = 2)

cowplot::plot_grid(
  plot_preliminary(d.p2, 3, 'expansive'), 
  p1, nrow = 2)

cowplot::plot_grid(
  plot_preliminary(d.p2, 5, 'expansive'), 
  p1, nrow = 2)

cowplot::plot_grid(
  plot_preliminary(d.p2, 6, 'expansive'), 
  p1, nrow = 2)

cowplot::plot_grid(
  plot_preliminary(d.p2, 7, 'expansive'), 
  p1, nrow = 2)

cowplot::plot_grid(
  plot_preliminary(d.p2, 7, c('prior', 'expansive')), 
  p1, nrow = 2)

We want to compute the posterior, which is the product of the prior and likelihood:

cowplot::plot_grid(
  plot_preliminary(d.p2, 7, c('prior', 'expansive'), linewidth_in = 0), 
  p1, nrow = 2)

cowplot::plot_grid(
  plot_preliminary(d.p2, 7, c('posterior')), 
  p1, nrow = 2)

Model 1: Equal standard deviations

Understanding the data

We plot the data distribution, the empirical density curve (blue line), and the theoretical density curve (black line).

pose_df %>%
  mutate(c = as.factor(condition)) %>%
  ggplot(aes(x = change)) +
  # data distribution
  geom_histogram(
    aes(y = ..density..),
    binwidth = 10,
    fill = theme_yellow,
    alpha = .75,
    color = 'white'
  ) +
  # empirical density curve
  geom_density(size = 1,
               adjust = 1.5,
               color = theme_blue) +
  #  theoretical density curve, a t distribition with mean = 16, sd = 19, and nu = 6
   geom_function(
    color = "#222222",
    linetype = 'dashed',
    fun = function(x)
      dstudent_t(x, mu = 16,  sigma = 39, df = 6),
    size = 1
  ) + 
  scale_x_continuous(limits = c(-200, 200)) +
  theme(
    axis.line.y = element_blank(),
    axis.text.y = element_blank(),
    axis.ticks.y = element_blank(),
    axis.title.y = element_blank(),
    axis.text.x = element_text(size = 20),
    axis.title.x = element_text(size = 24)
  )

Step 1: model specification

Here we use a mathematical expression for the model above.

\[ \begin{align} y_{i} &\sim \mathrm{Student\_t}(\mu, \sigma_{0}, \nu_{0}) \\ \mu &= \beta_{0} + \beta_{1} * x_i \\ \sigma_{0} &\sim \mathrm{HalfNormal}(0, 10) \\ \beta_{0} &\sim \mathrm{?} \\ \beta_{1} &\sim \mathrm{Normal}(0,2) \\ \nu_{0} &\sim \mathrm{?} \\ i & \in \{\mathrm{expansive}, \mathrm{constrictive}\} \end{align} \]

We translate this thought into brms formula using the function bf.

model.1.formula <- bf(
                      # we think change is affected by different conditions.
                      change ~ condition,
                      # to tell brms which response distribution to use
                      family = student()
                    )

Now we can use get_prior() to inspect the available priors and formula. As what we learned, we have priors for

tibble(get_prior(model.1.formula, pose_df))
prior class coef group resp dpar nlpar lb ub source
b default
b conditionexpansive default
student_t(3, 22.6, 38.8) Intercept default
gamma(2, 0.1) nu 1 default
student_t(3, 0, 38.8) sigma 0 default

prior checks

We look at the default priors from brms.

cowplot::plot_grid(
tibble(x = qstudent_t(ppoints(n = 500), df = 3, mu = 22.6, sigma = 38.8)) %>% 
  ggplot() +
  geom_density(aes(x = x), fill = theme_yellow, color = NA) +
  ggtitle('student_t(3, 22.6, 38.8) for Intercept') +
  coord_cartesian(xlim = c(-300, 300),expand = c(0)),
tibble(x = qgamma(ppoints(n = 500), shape = 2, rate = .1)) %>% 
  ggplot() +
  geom_density(aes(x = x), fill = theme_yellow, color = NA) +
  ggtitle('Gamma(2, 0.1) for nu') +
  coord_cartesian(xlim = c(1, 100), expand = 0),
tibble(x = qstudent_t(ppoints(n = 500), df = 3, mu = 0, sigma = 38.8)) %>% 
  ggplot() +
  geom_density(aes(x = x), fill = theme_yellow, color = NA) +
  ggtitle('student_t(3, 0, 38.8) for sigma') +
  coord_cartesian(xlim = c(0, 400),expand = c(0)),
ncol = 3)

We do a prior check for default priors to see the range of prior predictions.

model.1.checks_default <- brm(
  model.1.formula,
  data = pose_df, 
  family = student_t(),
  prior = c(
    # need proper priors for prior predictive checks
    prior(normal(0, 3), class = 'b')
  ),
  # to allow draw from prior distributions
  sample_prior = 'only',
  backend = BRM_BACKEND,
  # save the model
  file = 'rds/model.1.checks_default.rds',
  file_refit = 'on_change' 
)

We draw from prior distributions using predicted_draws.

model.1.defaultpriorsamples <- 
  model.1.checks_default %>% 
    predicted_draws(tibble(condition = c('expansive', 'constrictive')))

Take a look at the prior prediction draws. You can ignore .row,.chain, ‘.iteration’. The .draw is the ID for each draw. .prediction is the value we care about.

head(model.1.defaultpriorsamples)
condition .row .chain .iteration .draw .prediction
expansive 1 NA NA 1 71.188604
expansive 1 NA NA 2 155.807150
expansive 1 NA NA 3 9.000583
expansive 1 NA NA 4 75.183570
expansive 1 NA NA 5 18.108887
expansive 1 NA NA 6 -48.676288

We plot out the prediction draws. They look reasonable but have a wide range. We would expect so given the ranges of the priors.

model.1.defaultpriorsamples %>% 
  ggplot(aes(x = .prediction, group = condition)) +
  geom_density(alpha = .5, color = NA, adjust = 2, fill = theme_yellow) +
  theme_density_x + 
  ggtitle('Checks default priors of model.1')

We want to use narrow priors.

cowplot::plot_grid(
tibble(x = qstudent_t(ppoints(n = 1000), mu = 22.6, sigma = 10, df = 3)) %>% 
  ggplot() +
  geom_density(aes(x = x), fill = theme_yellow, color = NA) +
  ggtitle('t(3, 22.6, 10) for Intercept') +
  coord_cartesian(xlim = c(-30, 80), expand = 0),
tibble(x = qnorm(ppoints(n = 1000), mean = 0, sd = 2)) %>% 
  ggplot() +
  geom_density(aes(x = x), fill = theme_yellow, color = NA) +
  ggtitle('Normal(0, 2) for b') +
  coord_cartesian(xlim = c(-8, 8), expand = 0),
tibble(x = qnorm(ppoints(n = 1000), mean = 0, sd = 10)) %>% 
  ggplot() +
  geom_density(aes(x = x), fill = theme_yellow, color = NA) +
  ggtitle('HalfNormal(0, 10) for sigma') +
  coord_cartesian(xlim = c(0, 50), expand = c(0)),
ncol = 3)

We can do another prior check.

model.1.checks <- brm(
  model.1.formula,
  data = pose_df, 
  family = student_t(),
  prior = c(
    prior(student_t(3, 22.6, 10), class = "Intercept"),
    prior(normal(0, 2), class = 'b'),
    prior(normal(0, 10), class = 'sigma', lb = 0)
  ),
  sample_prior = 'only',
  backend = BRM_BACKEND,
  file = 'rds/model.1.checks.rds',
  file_refit = 'on_change' 
)

We draw from the new prior distribution.

model.1.priorsamples <- 
  model.1.checks %>% 
    predicted_draws(tibble(condition = c('expansive', 'constrictive')))

We compare two sets of prior distributions. Visually, using our priors have a narrower range.

cowplot::plot_grid(
  
model.1.defaultpriorsamples %>% 
  ggplot(aes(x = .prediction, group = condition)) +
  geom_density(alpha = .5, color = NA, adjust = 2, fill = theme_yellow) +
  theme_density_x + 
  # coord_cartesian(xlim = c(-800, 1000)) +
  ggtitle('Checks for default priors of model.1')
,
model.1.priorsamples %>% 
  ggplot(aes(x = .prediction, group = condition)) +
  geom_density(alpha = .5, color = NA, adjust = 2, fill = theme_yellow) +
  theme_density_x + 
  # coord_cartesian(xlim = c(-800, 1000)) +
  ggtitle('Checks for our priors of model.1')
,
ncol = 1)

Step 2: model fitting

Now we can fit the model. This model takes a

model.1 <- brm(
  model.1.formula,
  data = pose_df, 
  family = student_t(),
  prior = c(
    prior(normal(0, 2), class = 'b'),
    prior(normal(0, 10), class = 'sigma', lb = 0)
  ),
  backend = BRM_BACKEND,
  file = 'rds/model.1.rds'
)

Step 3: check posteriors

aspect 1: mcmc traces

First, we check the MCMC traces to ensure that the chains are mixed well.

color_scheme_set("teal")
mcmc_trace(model.1, facet_args = list(ncol = 4))

plot(model.1)

aspect 2: model metrics

We also check Rhat and ESS. We want Rhat to be close to 1 to ensure model convergence. We want Bulk ESS (effective sample size) to be at least a few hundreds (ideally, this should be at least a thousand.) to ensure reliable estimate of mean. We also want Tail ESS to be at this level. If ESSs are too low, it means there are too many correlations in posteriors draws. You need to increase the number of iterations and perhaps the number of chains.

summary(model.1)
##  Family: student 
##   Links: mu = identity; sigma = identity; nu = identity 
## Formula: change ~ condition 
##    Data: pose_df (Number of observations: 80) 
##   Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup draws = 4000
## 
## Population-Level Effects: 
##                    Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept             22.84      4.49    14.50    31.77 1.00     2686     2092
## conditionexpansive    -0.26      1.97    -4.10     3.62 1.00     2726     2987
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma    28.35      4.13    20.76    36.92 1.00     2180     2520
## nu        3.68      2.35     1.57     9.04 1.00     2172     2158
## 
## Draws were sampled using sample(hmc). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

aspect 3: visually

pose_df.bayesiant.y <- pose_df$change

pose_df.bayesiant.yrep <- posterior_predict(model.1, ndraws = 30, seed = 1234)

ppc_dens_overlay(pose_df.bayesiant.y, pose_df.bayesiant.yrep, linewidth = 2)

Now we check the posterior prediction of this model to ensure that it generates reasonable predictions.

model.1.predictions <- 
  model.1 %>% 
    predicted_draws(tibble(condition = c('expansive', 'constrictive')))
head(model.1.predictions)
condition .row .chain .iteration .draw .prediction
expansive 1 NA NA 1 22.785243
expansive 1 NA NA 2 -38.544134
expansive 1 NA NA 3 -2.368428
expansive 1 NA NA 4 23.209345
expansive 1 NA NA 5 -0.189108
expansive 1 NA NA 6 33.318285
plot_predictions <- function(model, df = NULL, title = ''){
  
  if(is.null(df))
      df = tibble(condition = c('expansive', 'constrictive'), 
                           participant = c(-1, -1))
  model %>% 
    predicted_draws(df,
                    seed = 1234,
                    ndraws = NULL,
                    allow_new_levels = TRUE,
                    sample_new_levels = 'uncertainty') %>% 
    ggplot(aes(x = .prediction, colour = condition), fill = NA) +
    geom_density(alpha = .5, size = 1, adjust = 2) +
    scale_color_theme() +
    theme_density_x + 
    scale_y_continuous(breaks = 0, labels = 'constrictive') + 
    scale_x_continuous(breaks = seq(-150, 250, by = 50)) +
    coord_cartesian(xlim = c(-150, 250)) + 
    ggtitle(paste0('Posterior predictions of ', deparse(substitute(model))))
}
cowplot::plot_grid(plot_predictions(model.1), 
                   p1 + 
                    scale_x_continuous(breaks = seq(-150, 250, by = 100)) +
                    coord_cartesian(xlim = c(-150, 250)), 
                   nrow = 2)

We now generate the posterior predictions of means, which are of interests here.

model.1.posteriors <- 
  model.1 %>% 
    epred_draws(tibble(condition = c('expansive', 'constrictive')))
plot_posteriors <- function(model, df = NULL, title = ''){
  
  if(is.null(df))
     df = tibble(condition = c('expansive', 'constrictive'))
  
  model %>% 
    epred_draws(df, 
                # ignoring random effects if there is any
                #seed = 1234,
                ndraws = NULL,
                re_formula = NA) %>% 
    ggplot(aes(x = .epred, fill = condition)) +
    geom_density(alpha = .5, size = 1, adjust = 2, color = NA) +
    scale_color_theme() +
    theme_density_x + 
    scale_y_continuous(breaks = 0, labels = 'constrictive') + 
    scale_x_continuous(limits = c(-150, 250), breaks = seq(-150, 250, by = 50)) +
    ggtitle(paste0('Posterior means of ', deparse(substitute(model))))
}

cowplot::plot_grid(plot_posteriors(model.1), p1, nrow = 2)

Model 2: the BEST test model

Step 1: model specification

This model is the BEST test model as described by Kruschke in the paper Bayesian estimation supersedes the t-test. In this model, \(\beta\) indicates the mean difference in the outcome variable between the two groups (in this case, the percent change in the BART scores). We fit different priors on \(\beta\) and set different weights on these priors to obtain our posterior estimate.

\[ \begin{align} y_{i} &\sim \mathrm{T}(\nu, \mu, \sigma) \\ \mu &= \beta_{0} + \beta_{1} * x_i \\ \sigma &= \sigma_{a} + \sigma_{b}*x_i \\ \beta_{1} &\sim \mathrm{Normal}(\mu_{0}, \sigma_{0}) \\ \sigma_a, \sigma_b &\sim \mathrm{Cauchy}(0, 2) \\ \nu &\sim \mathrm{exp}(1/30)\\ i & \in \{\mathrm{expansive}, \mathrm{constrictive}\} \end{align} \]

model.2.formula <- bf(# we think change is affected by different conditions.
                      change ~ condition,
                      sigma ~ condition,
                      # to tell brms which response distribution to use
                      family = student())
tibble(get_prior(model.2.formula, pose_df))
prior class coef group resp dpar nlpar lb ub source
b default
b conditionexpansive default
student_t(3, 22.6, 38.8) Intercept default
gamma(2, 0.1) nu 1 default
b sigma default
b conditionexpansive sigma default
student_t(3, 0, 2.5) Intercept sigma default

prior check

cowplot::plot_grid(
  tibble(x = qnorm(ppoints(n = 1000),  mean = 0, sd = 2)) %>% 
  ggplot() +
  geom_density(aes(x = x), fill = theme_yellow, color = NA) +
  ggtitle('Normal(0 ,2) for Intercept') +
  coord_cartesian(expand = c(0)),
tibble(x = qexp(ppoints(n = 1000), rate = 0.0333)) %>% 
  ggplot() +
  geom_density(aes(x = x), fill = theme_yellow, color = NA) +
  ggtitle('exponential(0.0333) for nu') +
  coord_cartesian(expand = 0),
tibble(x = qcauchy(ppoints(n = 1000), location = 0, scale = 2)) %>% 
  ggplot() +
  geom_density(aes(x = x), fill = theme_yellow, color = NA) +
  ggtitle('cauchy(0, 2) for sigma') +
  coord_cartesian(expand = c(0)),
ncol = 3)

model.2.priorchecks <- brm(
  model.2.formula,
  data = pose_df, 
  family = student_t(),
  prior = c(
    prior(normal(0, 2), class = 'b'),
    prior(cauchy(0, 2), class = 'b', dpar = 'sigma'),
    prior(exponential(0.0333), class = 'nu')
  ),
  sample_prior = 'only',
  backend = BRM_BACKEND,
  file = 'rds/model.2.priorchecks.rds',
  file_refit = 'on_change' 
)
model.2.priorchecks %>% 
 epred_draws(tibble(condition = c('expansive', 'constrictive'))) %>% 
  ggplot(aes(x = .epred, group = condition)) +
  geom_density(alpha = .5, color = NA, adjust = 2, fill = theme_yellow) +
  theme_density_x + 
  ggtitle('Checks priors of model.2')

Step 2: model fiting

model.2 <- brm(
  model.2.formula,
  data = pose_df, 
  family = student_t(),
  prior = c(
    prior(normal(0, 2), class = 'b'),
    prior(cauchy(0, 2), class = 'b', dpar = 'sigma'),
    prior(exponential(0.0333), class = 'nu')
  ),
  backend = BRM_BACKEND,
  file = 'rds/model.2.rds',
  file_refit = 'on_change' 
)

Step 3: posterior checks

summary(model.2)
##  Family: student 
##   Links: mu = identity; sigma = log; nu = identity 
## Formula: change ~ condition 
##          sigma ~ condition
##    Data: pose_df (Number of observations: 80) 
##   Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup draws = 4000
## 
## Population-Level Effects: 
##                          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## Intercept                   24.52      4.88    15.67    34.84 1.00     2333
## sigma_Intercept              3.41      0.20     3.00     3.79 1.00     2215
## conditionexpansive          -0.09      1.96    -3.93     3.64 1.00     3639
## sigma_conditionexpansive     0.15      0.21    -0.27     0.57 1.00     3051
##                          Tail_ESS
## Intercept                    2276
## sigma_Intercept              2420
## conditionexpansive           2377
## sigma_conditionexpansive     2668
## 
## Family Specific Parameters: 
##    Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## nu     6.03      8.24     1.76    24.62 1.00     1860     1782
## 
## Draws were sampled using sample(hmc). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
wrap_plots(plot_predictions(model.2) + scale_x_continuous(limits = c(-200,250), expand = c(0,0), breaks = seq(-150, 250, by = 50)) 
                   , 
                   p1 + scale_x_continuous(limits = c(-200,250), expand = c(0,0), breaks = seq(-150, 250, by = 50)) , 
                   nrow = 2)

wrap_plots(
  plot_posteriors(model.2), 
  p1,
nrow = 2)

model.2.posteriors <- model.2 %>% 
                epred_draws(tibble(condition = c('expansive', 'constrictive')), 
                #seed = 1234,
                ndraws = NULL,
                re_formula = NA) 

Model 3: the BEST model with better intercepts

Step 1: model specification

\[ \begin{align} y_{i} &\sim \mathrm{T}(\nu, \mu, \sigma) \\ \mu &= \beta_{i,j} + \beta_{1} * x_i \\ \sigma &= \sigma_{a} + \sigma_{b}*x_i \\ \beta_{1} &\sim \mathrm{Normal}(0, 2) \\ \sigma_a, \sigma_b &\sim \mathrm{HalfNormal}(0, 10) \\ \nu &\sim \mathrm{exp}(1/30)\\ i & \in \{\mathrm{expansive}, \mathrm{constrictive}\}\\ j & \in \{1, ..., \mathrm{N}\} \end{align} \]

model.3.formula <- bf(# we think change is affected by different conditions.
                      change ~ condition + (1|participant),
                      sigma ~ condition,
                      # to tell brms which response distribution to use
                      family = student())
tibble(get_prior(model.3.formula, pose_df))
prior class coef group resp dpar nlpar lb ub source
b default
b conditionexpansive default
student_t(3, 22.6, 38.8) Intercept default
gamma(2, 0.1) nu 1 default
student_t(3, 0, 38.8) sd 0 default
sd participant default
sd Intercept participant default
b sigma default
b conditionexpansive sigma default
student_t(3, 0, 2.5) Intercept sigma default

Step 2: model fiting

model.3 <- brm(
  model.3.formula,
  data = pose_df, 
  family = student_t(),
  prior = c(
    prior(normal(0, 2), class = 'b'),
    prior(normal(0, 2), class = 'b', dpar = 'sigma'),
    prior(normal(0, 2), class = 'sd', group = 'participant', lb = 0),
    prior(exponential(0.0333), class = 'nu')
  ),
  #backend = BRM_BACKEND,
  file = 'rds/model.3.rds',
  file_refit = 'on_change' 
)

Step 3: posterior checks

summary(model.3)
##  Family: student 
##   Links: mu = identity; sigma = log; nu = identity 
## Formula: change ~ condition + (1 | participant) 
##          sigma ~ condition
##    Data: pose_df (Number of observations: 80) 
##   Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup draws = 4000
## 
## Group-Level Effects: 
## ~participant (Number of levels: 80) 
##               Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sd(Intercept)     1.63      1.23     0.06     4.57 1.00     3650     2196
## 
## Population-Level Effects: 
##                          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## Intercept                   24.48      4.77    15.62    34.43 1.00     5853
## sigma_Intercept              3.40      0.20     3.01     3.79 1.00     5493
## conditionexpansive          -0.09      1.92    -3.77     3.73 1.00    12370
## sigma_conditionexpansive     0.15      0.21    -0.29     0.58 1.00     9195
##                          Tail_ESS
## Intercept                    2933
## sigma_Intercept              2867
## conditionexpansive           2502
## sigma_conditionexpansive     2684
## 
## Family Specific Parameters: 
##    Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## nu     6.09      8.76     1.78    24.32 1.00     4880     2600
## 
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
cowplot::plot_grid(plot_predictions(model.3), p1, nrow = 2)

cowplot::plot_grid(plot_posteriors(model.3), p1, nrow = 2)

model.3.posteriors <-
  model.3 %>% 
  epred_draws(tibble(condition = c('expansive', 'constrictive')),
               re_formula = NA)
wrap_plots(
model.3.posteriors %>%
  mutate(model = 'model 3') %>%
  rbind(
    model.2.posteriors %>% 
      mutate(model = 'model 2'))  %>% 
  rbind(
    model.1.posteriors %>% 
      mutate(model = 'model 1')) %>% 
ggplot() + 
  geom_density(aes(x = .epred, fill = condition), adjust = 1.5, color = NA, alpha = .5) +
  facet_grid(model ~ .) +
  scale_x_continuous(limits = c(-150, 250), breaks = seq(-150, 250, by = 50)) +
  scale_color_theme() +
  theme_density_x +
  ggtitle('Compare the means of all three models'),
p1, nrow = 2, heights = c(4,1.5))

Model 4: Negative-Binomial Regression model

Understanding the data

pose_raw_df = read.csv("data/posture_data-raw.csv") %>%
  mutate(participant = factor(participant)) %>%
  rename(trial = trial.number)

head(pose_raw_df)
participant condition total.money trial trial.money exploded pumps life
1 expansive 910 0 62 0 62 72
1 expansive 910 1 0 1 27 27
1 expansive 910 2 47 0 47 98
1 expansive 910 3 0 1 86 86
1 expansive 910 4 60 0 60 104
1 expansive 910 5 0 1 26 26
tibble(x = seq(1, 160, by = 1)) %>%
  ggplot(aes(x)) +
  geom_function(
    color = theme_blue,
    linetype = 'dashed',
    fun = function(x) # need to fix
      dnbinom(round(x), size = 5, mu = 64),
    size = 1
  ) +
  # stat_function(geom = "area", fill = theme_red, fun = function(x) dnbinom(round(x), size = 5, mu = 64), xlim = c(128, 160), alpha = 0.5) +
  geom_vline(xintercept = 0, color = "red", size = 1, alpha = 0.5) +
  geom_vline(xintercept = 128, color = "red", size = 1, alpha = 0.5) +
  coord_cartesian(xlim = c(0, 160)) +
  theme(
    axis.line.y = element_blank(),
    axis.text.y = element_blank(),
    axis.ticks.y = element_blank(),
    axis.title.y = element_blank(),
    axis.text.x = element_text(size = 16),
    axis.title.x = element_blank()
  )

pose_raw_df %>%
  mutate(c = as.factor(condition)) %>%
  ggplot(aes(x = pumps)) +
  geom_histogram(
    aes(y = ..density..),
    binwidth = 2,
    fill = theme_yellow,
    alpha = .5,
    color = 'white'
  ) +
  geom_density(size = 1,
               adjust = 3,
               color = theme_blue) +
  geom_function(
    color = "#222222",
    linetype = 'dashed',
    fun = function(x) # need to fix
      dnbinom(round(x), size = 3, mu = 40), # this does not use the log-link but brm does so the prior below is # log(387)
    size = 1
  ) +
  theme(
    axis.line.y = element_blank(),
    axis.text.y = element_blank(),
    axis.ticks.y = element_blank(),
    axis.title.y = element_blank()
  )

Step 1: model specification

\[ \begin{align} y_{i} &\sim \mathrm{Neg-Binomial}(\mu, \phi) \\ log(\mu) &= \beta_{0} + \beta_{1} . x_i \\ \beta_{0} &\sim \mathrm{Normal}(4.1, 0.5) \\ \beta_{0, j} &\sim \mathrm{Student\_t}(3, 0, 1) \\ \beta_{1} &\sim \mathrm{Normal}(0, 0.5) \\ \phi &\sim \mathrm{Gamma}(5, 1) \\ i & \in \{\mathrm{expansive}, \mathrm{constrictive}\}\\ \end{align} \]

Step 2: prior predictive checks

model.4.prior_pred = brm(pumps ~ 1 + condition,
                      data = pose_raw_df, family = negbinomial(),
                      prior = c(prior(normal(4.1, 0.5), class = Intercept),
                                prior(normal(0, 0.5), class = b),
                                prior(gamma(5, 1), class = shape)
                      ),
                      backend = BRM_BACKEND,
                      file = 'rds/model.4.prior_predictive.rds',
                      file_refit = 'on_change' ,
                      sample_prior = "only",
                      iter = 4000, warmup = 1000, cores = 4, chains = 4)
model.4.prior_pred.samples <- model.4.prior_pred %>% 
    predicted_draws(tibble(condition = c('expansive', 'constrictive')), re_formula = NA)

head(model.4.prior_pred.samples)
condition .row .chain .iteration .draw .prediction
expansive 1 NA NA 1 99
expansive 1 NA NA 2 113
expansive 1 NA NA 3 19
expansive 1 NA NA 4 66
expansive 1 NA NA 5 30
expansive 1 NA NA 6 56
ggplot() +
geom_histogram(
  data = pose_raw_df,
  mapping = aes(x = pumps, y = ..density..),
  binwidth = 10,
  alpha = 0.5,
  fill = theme_yellow,
  color = 'white'
) +
geom_density(model.4.prior_pred.samples,
               mapping = aes(x = .prediction, y = ..density..), adjust = 1, stroke = theme_blue, size = 1) +
theme_density_x + 
coord_cartesian(xlim = c(0, 150)) +
theme(
  axis.text.x = element_text(size = 16)
)

Step 3: model fitting

model.4 = brm(pumps ~ 1 + condition,
                      data = pose_raw_df, family = negbinomial(),
                      prior = c(prior(normal(4.1, 0.5), class = Intercept),
                                prior(normal(0, 0.5), class = b),
                                prior(gamma(5, 1), class = shape)
                      ),
                      backend = BRM_BACKEND,
                      file = 'rds/model.4.rds',
                      file_refit = 'on_change' ,
                      iter = 4000, warmup = 1000, cores = 4, chains = 4)
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summary(model.4)
##  Family: negbinomial 
##   Links: mu = log; shape = identity 
## Formula: pumps ~ 1 + condition 
##    Data: pose_raw_df (Number of observations: 2400) 
##   Draws: 4 chains, each with iter = 4000; warmup = 1000; thin = 1;
##          total post-warmup draws = 12000
## 
## Population-Level Effects: 
##                    Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept              3.64      0.01     3.61     3.66 1.00    11789     9145
## conditionexpansive    -0.01      0.02    -0.05     0.03 1.00    10845     8766
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## shape     4.59      0.15     4.32     4.89 1.00    10138     8289
## 
## Draws were sampled using sample(hmc). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
pose_raw_df.bayesian_poisson.y <- pose_raw_df$pumps

pose_raw_df.bayesian_poisson.yrep <- posterior_predict(model.4, ndraws = 30, seed = 1234)

ppc_dens_overlay(y = pose_raw_df.bayesian_poisson.y,
                 yrep = pose_raw_df.bayesian_poisson.yrep)

Step 4: model interpretation

The results of this model are on the log-odds scale. What do the coefficients mean? The simplest way is to simply transform the data into a more interpretable scale and visualise the results:

p.model.4 = pose_raw_df %>%
  ggplot() +
  geom_point(aes(x = pumps, y = condition, colour = condition), 
             position = position_jitter(height = 0.1), alpha = 0.7) +
  scale_color_theme() + 
  labs(y = "Condition") +
  theme(
    legend.position = "none", 
    axis.line.y = element_blank(), 
    axis.ticks.y = element_blank(),
    axis.title.y = element_blank()
  ) +
  scale_x_continuous(limits = c(0, 130), breaks = seq(0, 150, by = 30))

draws.model.4 = plot_predictions(model.4, df = crossing(condition = c('expansive', 'constrictive'), 
                                          trial = 0:29,
                                          participant = 0)) + 
    scale_x_continuous(breaks = seq(0, 150, by = 30)) +
    coord_cartesian(xlim = c(0, 150))

wrap_plots(
  draws.model.4, 
  p.model.4,
nrow = 2)

We generate the average of 30 trials using an average participant.

model.4.posteriors <-  model.4 %>% 
    epred_draws(crossing(condition = c('expansive', 'constrictive'),     
                       trial = 0:29),
                re_formula = NA,
                allow_new_levels  = FALSE) %>% 
  mutate(model = 'model 4')  %>% 
  group_by(.draw, condition) %>% 
  summarise(.epred = mean(.epred))
model.4.posteriors %>%
  ungroup() %>%
  compare_levels(.epred, by = condition) %>%
  median_qi(.width = .95) %>%
  ggplot() + 
  geom_pointinterval(aes(x = .epred, y = condition, xmin = .lower, xmax = .upper)) +
  geom_vline(xintercept = 0, linetype = 2, color = "#979797") +
  scale_x_continuous(breaks = seq(-10, 10, by = 2)) +
  coord_cartesian(xlim = c(-10, 10)) +
  xlab("Difference in Mean") +
  theme(
    axis.text.x = element_text(size = 14),
    axis.text.y = element_text(size = 14),
    axis.title.x = element_text(size = 16),
    axis.title.y = element_blank()
  )

Plot out the posteriors for mean.

model.4.posteriors %>% 
 ggplot(aes(x = .epred, fill = condition)) +
    stat_halfeye(.width = .95) +
    scale_color_theme() +
    facet_wrap(condition ~ ., ncol = 1) + 
    theme_density_x + 
    scale_x_continuous(limits = c(30, 50), breaks = seq(0, 60, by = 10)) +
    ggtitle(paste0('Posterior means of ', deparse(substitute(model))))

From this, we can see that there does not appear to be a difference between the two conditions.

Reporting

Let’s use model 4

model

We will need the ESSs and RHat from this print. Also, if you want to report the standard deviation of random intercepts or CIs for other paramaters. You can find them via summary(..)

summary(model.4)
##  Family: negbinomial 
##   Links: mu = log; shape = identity 
## Formula: pumps ~ 1 + condition 
##    Data: pose_raw_df (Number of observations: 2400) 
##   Draws: 4 chains, each with iter = 4000; warmup = 1000; thin = 1;
##          total post-warmup draws = 12000
## 
## Population-Level Effects: 
##                    Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept              3.64      0.01     3.61     3.66 1.00    11789     9145
## conditionexpansive    -0.01      0.02    -0.05     0.03 1.00    10845     8766
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## shape     4.59      0.15     4.32     4.89 1.00    10138     8289
## 
## Draws were sampled using sample(hmc). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Credible Intervals

We compute the credible intervals (CIs; Bayesian analogy to confidence intervals) for the two conditions.

model.4.posteriors.CI <- 
model.4.posteriors %>% 
  group_by(condition) %>% 
  median_qi(.epred, width = .95)
model.4.posteriors.CI
condition .epred .epred.lower .epred.upper width width.lower width.upper .width .point .interval
constrictive 37.94395 36.92430 39.03394 0.95 0.95 0.95 0.95 median qi
expansive 37.67453 36.64203 38.74556 0.95 0.95 0.95 0.95 median qi
model.4.posteriors %>% 
ggplot() +
  geom_density(aes(x = .epred, fill = condition), alpha = .5, color = NA) +
  geom_point(model.4.posteriors.CI, 
             mapping = aes(x = .epred, y = 0), size = 3) + 
  geom_errorbarh(model.4.posteriors.CI, 
                 mapping = aes(x = .epred, xmin = .epred.lower, xmax = .epred.upper, y = 0), height = 0, linewidth = 1.5) + 
  facet_wrap(condition ~ ., ncol = 1) + 
  scale_x_continuous(limits = c(0, 80)) + 
  scale_y_continuous(expand = c(.02,.02)) + 
  scale_color_theme() +
  theme_density_x 

subtraction

model.4.posteriors_diff <- 
model.4.posteriors %>% 
  ungroup() %>% 
  compare_levels(variable = .epred, by = condition) %>% 
  ungroup()
head(model.4.posteriors_diff)
.draw condition .epred
1 expansive - constrictive 1.8014050
2 expansive - constrictive 0.4099562
3 expansive - constrictive -0.1681858
4 expansive - constrictive 1.5619406
5 expansive - constrictive 0.0938069
6 expansive - constrictive -0.3691244
model.4.posteriors_diff.CI <-
model.4.posteriors_diff %>% 
  median_qi(.epred)

model.4.posteriors_diff.CI
.epred .lower .upper .width .point .interval
-0.2769801 -1.778706 1.204555 0.95 median qi
model.4.posteriors_diff %>% 
ggplot() +
  geom_density(aes(x = .epred), alpha = .5,  fill = 'skyblue', color = NA, adjust = 2) +
  geom_point(model.4.posteriors_diff.CI, 
             mapping = aes(x = .epred, y = 0), size = 3) + 
  geom_errorbarh(model.4.posteriors_diff.CI, 
                 mapping = aes(x = .epred, xmin = .lower, xmax = .upper, y = 0), height = 0, linewidth = 1.5) + 
  #scale_x_continuous(limits = c(-50, 50)) + 
  xlab('expansive - constrictive') +
  ggtitle('Mean difference in expansive and constrictive') + 
  geom_vline(xintercept = 0, linetype = 2) + 
  scale_y_continuous(expand = c(.02,.02)) + 
  scale_x_continuous(limits = c(-10, 10), breaks = seq(-10, 10, by = 5)) + 
  scale_color_theme() +
  theme_density_x 

Model 5 (Bonus): Negative-Binomial Regression model with other predictors

Step 1: model specification

\[ \begin{align} y_{i} &\sim \mathrm{Neg-Binomial}(\mu, \phi) \\ log(\mu) &= \beta_{0,k} + \beta_{1} . x_i + \beta_j . x_j \\ \beta_{0, j} &\sim \mathrm{Normal}(\alpha, \sigma) \\ \alpha &\sim \mathrm{Normal}(4.1, 0.5) \\ \sigma &\sim \mathrm{Exponential}(2) \\ \beta_{0, j} &\sim \mathrm{Student\_t}(3, 0, 1) \\ \beta_{1} &\sim \mathrm{Normal}(0, 0.5) \\ \phi &\sim \mathrm{Gamma}(5, 1) \\ i & \in \{\mathrm{expansive}, \mathrm{constrictive}\}\\ j & \in \{1, ..., \mathrm{N}\} \end{align} \]

Step 2: model fitting

model.5 = brm(pumps ~ 1 + condition + trial + (1|participant),
                      data = pose_raw_df, family = negbinomial(),
                      prior = c(prior(normal(4.1, 0.5), class = Intercept),
                                prior(normal(0, 0.5), class = b),
                                prior(exponential(2), class = sd),
                                prior(gamma(5, 1), class = shape)
                      ),
                      backend = BRM_BACKEND,
                      file = 'rds/model.4.rds',
                      file_refit = 'on_change' ,
                      iter = 4000, warmup = 1000, cores = 4, chains = 4)
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summary(model.5)
##  Family: negbinomial 
##   Links: mu = log; shape = identity 
## Formula: pumps ~ 1 + condition + trial + (1 | participant) 
##    Data: pose_raw_df (Number of observations: 2400) 
##   Draws: 4 chains, each with iter = 4000; warmup = 1000; thin = 1;
##          total post-warmup draws = 12000
## 
## Group-Level Effects: 
## ~participant (Number of levels: 80) 
##               Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sd(Intercept)     0.31      0.03     0.27     0.37 1.00     1396     2864
## 
## Population-Level Effects: 
##                    Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept              3.52      0.05     3.42     3.62 1.00      855     1978
## conditionexpansive    -0.02      0.07    -0.16     0.12 1.00      760     1717
## trial                  0.01      0.00     0.00     0.01 1.00    11968     7407
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## shape     7.42      0.25     6.93     7.94 1.00    12184     9233
## 
## Draws were sampled using sample(hmc). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
pose_raw_df.bayesian_poisson.y <- pose_raw_df$pumps

pose_raw_df.bayesian_poisson.yrep <- posterior_predict(model.5, ndraws = 30, seed = 1234)

ppc_dens_overlay(y = pose_raw_df.bayesian_poisson.y,
                 yrep = pose_raw_df.bayesian_poisson.yrep)

summary(model.5)
##  Family: negbinomial 
##   Links: mu = log; shape = identity 
## Formula: pumps ~ 1 + condition + trial + (1 | participant) 
##    Data: pose_raw_df (Number of observations: 2400) 
##   Draws: 4 chains, each with iter = 4000; warmup = 1000; thin = 1;
##          total post-warmup draws = 12000
## 
## Group-Level Effects: 
## ~participant (Number of levels: 80) 
##               Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sd(Intercept)     0.31      0.03     0.27     0.37 1.00     1396     2864
## 
## Population-Level Effects: 
##                    Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept              3.52      0.05     3.42     3.62 1.00      855     1978
## conditionexpansive    -0.02      0.07    -0.16     0.12 1.00      760     1717
## trial                  0.01      0.00     0.00     0.01 1.00    11968     7407
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## shape     7.42      0.25     6.93     7.94 1.00    12184     9233
## 
## Draws were sampled using sample(hmc). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Step 3: model interpretation

The results of this model are on the log-odds scale. What do the coefficients mean? The simplest way is to simply transform the data into a more interpretable scale and visualise the results:

p.model.5 = pose_raw_df %>%
  ggplot() +
  geom_point(aes(x = pumps, y = condition, colour = condition), 
             position = position_jitter(height = 0.1), alpha = 0.7) +
  scale_color_theme() + 
  labs(y = "Condition") +
  theme(
    legend.position = "none", 
    axis.line.y = element_blank(), 
    axis.ticks.y = element_blank(),
    axis.title.y = element_blank()
  ) +
  scale_x_continuous(limits = c(0, 130), breaks = seq(0, 150, by = 30))

draws.model.5 = plot_predictions(model.5, df = crossing(condition = c('expansive', 'constrictive'), 
                                          trial = 0:29,
                                          participant = 0)) + 
    scale_x_continuous(breaks = seq(0, 150, by = 30)) +
    coord_cartesian(xlim = c(0, 150))

wrap_plots(
  draws.model.5, 
  p.model.5,
nrow = 2)

We generate the average of 30 trials using an average participant.

model.5.posteriors <-  model.5 %>% 
    epred_draws(crossing(condition = c('expansive', 'constrictive'),     
                       trial = 0:29),
                re_formula = NA,
                allow_new_levels  = FALSE) %>% 
  mutate(model = 'model 5')  %>% 
  group_by(.draw, condition) %>% 
  summarise(.epred = mean(.epred))

model.5.posteriors %>%
  ungroup() %>%
  compare_levels(.epred, by = condition) %>%
  median_qi(.width = .95) %>%
  ggplot() + 
  geom_pointinterval(aes(x = .epred, y = condition, xmin = .lower, xmax = .upper)) +
  geom_vline(xintercept = 0, linetype = 2, color = "#979797") +
  scale_x_continuous(breaks = seq(-10, 10, by = 2)) +
  coord_cartesian(xlim = c(-10, 10)) +
  xlab("Difference in Mean") +
  theme(
    axis.text.x = element_text(size = 14),
    axis.text.y = element_text(size = 14),
    axis.title.x = element_text(size = 16),
    axis.title.y = element_blank()
  )

Plot out the posteriors for mean.

model.5.posteriors %>% 
 ggplot(aes(x = .epred, fill = condition)) +
    stat_halfeye(.width = .95) +
    scale_color_theme() +
    facet_wrap(condition ~ ., ncol = 1) + 
    theme_density_x + 
    scale_x_continuous(limits = c(30, 50), breaks = seq(0, 60, by = 10)) +
    ggtitle(paste0('Posterior means of ', deparse(substitute(model))))

From this, we can see that there does not appear to be a difference between the two conditions.

Session info

sessionInfo()
## R version 4.2.2 (2022-10-31)
## Platform: aarch64-apple-darwin20 (64-bit)
## Running under: macOS Ventura 13.2
## 
## Matrix products: default
## BLAS:   /Library/Frameworks/R.framework/Versions/4.2-arm64/Resources/lib/libRblas.0.dylib
## LAPACK: /Library/Frameworks/R.framework/Versions/4.2-arm64/Resources/lib/libRlapack.dylib
## 
## locale:
## [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
## 
## attached base packages:
## [1] stats     graphics  grDevices utils     datasets  methods   base     
## 
## other attached packages:
##  [1] cmdstanr_0.5.2      knitr_1.39          ggplot2_3.4.0      
##  [4] bayesplot_1.9.0     ggdist_3.1.1        tidybayes_3.0.2    
##  [7] brms_2.17.0         Rcpp_1.0.8.3        modelr_0.1.8       
## [10] broom.mixed_0.2.9.4 broom_1.0.0         patchwork_1.1.2    
## [13] gtools_3.9.2.1      forcats_0.5.1       tidyr_1.2.0        
## [16] purrr_0.3.4         tibble_3.1.7        dplyr_1.0.9        
## 
## loaded via a namespace (and not attached):
##   [1] colorspace_2.0-3     ellipsis_0.3.2       ggridges_0.5.3      
##   [4] markdown_1.1         base64enc_0.1-3      rstudioapi_0.13     
##   [7] listenv_0.8.0        furrr_0.3.1          farver_2.1.1        
##  [10] rstan_2.21.5         bit64_4.0.5          svUnit_1.0.6        
##  [13] DT_0.23              fansi_1.0.3          mvtnorm_1.1-3       
##  [16] diffobj_0.3.5        bridgesampling_1.1-2 codetools_0.2-18    
##  [19] splines_4.2.2        shinythemes_1.2.0    jsonlite_1.8.0      
##  [22] shiny_1.7.1          readr_2.1.2          compiler_4.2.2      
##  [25] backports_1.4.1      assertthat_0.2.1     Matrix_1.5-1        
##  [28] fastmap_1.1.0        cli_3.4.1            later_1.3.0         
##  [31] htmltools_0.5.2      prettyunits_1.1.1    tools_4.2.2         
##  [34] igraph_1.3.1         coda_0.19-4          gtable_0.3.0        
##  [37] glue_1.6.2           reshape2_1.4.4       posterior_1.2.1     
##  [40] jquerylib_0.1.4      vctrs_0.5.1          nlme_3.1-160        
##  [43] crosstalk_1.2.0      tensorA_0.36.2       xfun_0.31           
##  [46] stringr_1.4.0        globals_0.15.0       ps_1.7.0            
##  [49] mime_0.12            miniUI_0.1.1.1       lifecycle_1.0.3     
##  [52] future_1.26.1        zoo_1.8-10           scales_1.2.0        
##  [55] vroom_1.5.7          colourpicker_1.1.1   hms_1.1.1           
##  [58] promises_1.2.0.1     Brobdingnag_1.2-7    parallel_4.2.2      
##  [61] inline_0.3.19        shinystan_2.6.0      yaml_2.3.5          
##  [64] gridExtra_2.3        loo_2.5.1            StanHeaders_2.21.0-7
##  [67] sass_0.4.1           stringi_1.7.6        highr_0.9           
##  [70] dygraphs_1.1.1.6     checkmate_2.1.0      pkgbuild_1.3.1      
##  [73] rlang_1.0.6          pkgconfig_2.0.3      matrixStats_0.62.0  
##  [76] distributional_0.3.0 evaluate_0.15        lattice_0.20-45     
##  [79] labeling_0.4.2       rstantools_2.2.0     htmlwidgets_1.5.4   
##  [82] cowplot_1.1.1        bit_4.0.4            tidyselect_1.1.2    
##  [85] processx_3.5.3       parallelly_1.31.1    plyr_1.8.7          
##  [88] magrittr_2.0.3       R6_2.5.1             generics_0.1.3      
##  [91] DBI_1.1.2            pillar_1.7.0         withr_2.5.0         
##  [94] xts_0.12.1           abind_1.4-5          crayon_1.5.1        
##  [97] arrayhelpers_1.1-0   utf8_1.2.2           tzdb_0.3.0          
## [100] rmarkdown_2.14       grid_4.2.2           data.table_1.14.2   
## [103] callr_3.7.0          threejs_0.3.3        digest_0.6.29       
## [106] xtable_1.8-4         httpuv_1.6.5         RcppParallel_5.1.5  
## [109] stats4_4.2.2         munsell_0.5.0        bslib_0.3.1         
## [112] shinyjs_2.1.0